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IGPM490.pdf April 2019 
TITLE 
Trace finite element methods for surface vectorLaplace equations 
AUTHORS 
Thomas Jankuhn, Arnold Reusken 
ABSTRACT 
In this paper we analyze a class of trace finite element methods (TraceFEM) for
the discretization of vectorLaplace equations. A key issue in the finite element discretization of
such problems is the treatment of the constraint that the unknown vector field must be tangential
to the surface (“tangent condition”). We study three different natural techniques for treating the
tangent condition, namely a consistent penalty method, a simpler inconsistent penalty method and a
Lagrange multiplier method. A main goal of the paper is to present an analysis that reveals important
properties of these three different techniques for treating the tangent constraint. A detailed error
analysis is presented that takes the approximation of both the geometry of the surface and the
solution of the partial differential equation into account. Error bounds in the energy norm are
derived that show how the discretization error depends on relevant parameters such as the degree
of the polynomials used for the approximation of the solution, the degree of the polynomials used
for the approximation of the level set function that characterizes the surface, the penalty parameter
and the degree of the polynomials used for the approximation of Lagrange multiplier. 
KEYWORDS 
vectorLaplace, trace finite element method 